A more straightforward derivation of the LR for a database search

DNA database matches: A p versus np problem

The evidential value of a unique DNA database match has been extensively discussed. In principle the matter has been mathematically resolved, since the posterior odds on the match being with the trace donor are unambiguously defined. There are multiple ways to express these odds as a product of likelihood ratio and prior odds, and so the mathematics do not immediately tell us what to do in concrete cases, in particular which likelihood ratio to choose for reporting. With p the random match probability for the matching person, if innocent, and n the database size, both 1/p, originating from a suspect-centered framework, and 1/(np), originating from a database-centered framework, arise as likelihood ratio. Both have been defended and both have been criticized in the literature. We will clarify the situation by not introducing models and choices of prior probabilities until they are needed. This allows to derive the posterior odds in their most general form, which applies whenever we know that a single person among a list is not excluded as potential trace donor. We show that we need only three probabilities, that pertain to the observed match, to the database, and to the matching person respectively. How these required probabilities behave in a given context, then, differs from one situation to another. This is understandable since database searches may be done under various circumstances. They may be carried out with or without a suspect already in mind and, depending on the operational procedures, one may or may not be informed about the personal details of the person who gives the match. We show how to evaluate the required probabilities in all such cases. We will motivate why we believe that for some database searches, the 1/p likelihood ratio is more natural, whereas for others, 1/(np) seems the more sensible choice. This is not motivated by the mathematics: mathematically, the approaches are equivalent. It is motivated by considering which model best reflects the actual situation, taking into account what question was asked to begin with, and by the practical consideration of judging which likelihood ratio comes closer to the posterior odds based on the information available in the case. This article is intended to be both a research and a review article, and we end with an in-depth discussion of various arguments that have been brought forward in favor or against either 1/p or 1/(np).

DNA commission of the International society for forensic genetics: Assessing the value of forensic biological evidence - Guidelines highlighting the importance of propositions: Part I: evaluation of DNA profiling comparisons given (sub-) source propositions

The interpretation of evidence continues to be one of the biggest challenges facing the forensic community. This is the first of two papers intended to provide advice on difficult aspects of evaluation and in particular on the formulation of propositions. The scientist has a dual role: investigator (crime-focused), where often there is no suspect available and a database search may be required; evaluator (suspect-focused), where the strength of evidence is assessed in the context of the case. In investigative mode, generally the aim is to produce leads regarding the source of the DNA. Sub-source level propositions will be adequate to help identify potential suspects who can be further investigated by the authorities. Once in evaluative mode, given the defence version of events of the person of interest, it may become necessary to consider alternatives that go beyond the source of the DNA (i.e., to consider activity level propositions). In the evaluation phase, it is crucial that formulation of propositions allows the assessment of all the results that will help with the issue at hand. Propositions should therefore be precise (indication of the number of contributors, information on the relevant population etc.), be about causes, not effects (e.g. a 'matching' DNA profile) and to avoid bias, must not be findings-led. This means that ideally, propositions should be decided based on the case information and before the results of the comparisons are known. This paper primarily reflects upon what has been coined as "sub-source level propositions". These are restricted to the evaluation of the DNA profiles themselves, and help answer the issue regarding the source of the DNA. It is to be emphasised that likelihood ratios given sub-source level propositions cannot be carried over to a different level - for example, activity level propositions, where the DNA evidence is put into the context of the alleged activities. This would be highly misleading and could give rise to miscarriages of justice; this will be discussed in a second paper. The value of forensic results depends not only on propositions, but also on the type of results (e.g. allelic designations, peak heights, replicates) and upon the model used: it is therefore important to discuss these aspects. Finally, since communication is key to help understanding by courts, we will explore how to convey the value of the results and explain the importance of avoiding the practice of transposing the conditional.